The embedded homology of hypergraphs and applications

Bressan, Stéphane; Li, Jingyan; Ren, Shiquan; Wu, Jie

doi:10.4310/ajm.2019.v23.n3.a6

Cited by 36 publications

(46 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, traditional graphs only represent the pairwise relationships between entries. Therefore, hypergraphs, a generalization of graphs that describe the multi-way relationships of mathematical structures have been developed to capture the high-level complexity of data [2,6]. Mathematically, graphs and hypergraphs are intrinsically related to the simplicial complexes, which have broader use in computational topology.…”

Section: Introductionmentioning

confidence: 99%

Persistent spectral graph

Wang

Nguyen

Wei

2020

Numer Methods Biomed Eng

106

View full text Add to dashboard Cite

Persistent homology is constrained to purely topological persistence, while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low‐dimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high‐dimensional datasets. For a point‐cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, that is, the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non‐harmonic spectra of the Laplacian matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerene stability is predicted by using both harmonic spectra and non‐harmonic persistent spectra, while the latter spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B‐factors for which current popular biophysical models break down.

show abstract

Section: Introductionmentioning

confidence: 99%

Persistent spectral graph

Wang

Nguyen

Wei

2020

Numer Methods Biomed Eng

106

View full text Add to dashboard Cite

show abstract

“…The random walk hypergraph Laplacian is defined as L rw = 2I − D v −1 H H T . Recently, embedded homology, persistent homology, and weighted (Hodge) Laplacians have been developed for hypergraphs (37,38).…”

Section: Spectral Hypergraphmentioning

confidence: 99%

Persistent spectral–based machine learning (PerSpect ML) for protein-ligand binding affinity prediction

2021

View full text Add to dashboard Cite

Molecular descriptors are essential to not only quantitative structure-activity relationship (QSAR) models but also machine learning–based material, chemical, and biological data analysis. Here, we propose persistent spectral–based machine learning (PerSpect ML) models for drug design. Different from all previous spectral models, a filtration process is introduced to generate a sequence of spectral models at various different scales. PerSpect attributes are defined as the function of spectral variables over the filtration value. Molecular descriptors obtained from PerSpect attributes are combined with machine learning models for protein-ligand binding affinity prediction. Our results, for the three most commonly used databases including PDBbind-2007, PDBbind-2013, and PDBbind-2016, are better than all existing models, as far as we know. The proposed PerSpect theory provides a powerful feature engineering framework. PerSpect ML models demonstrate great potential to significantly improve the performance of learning models in molecular data analysis.

show abstract

“…There has been little application of the concepts of TDA to hypergraphs directly. However, there has been work some preliminary works connecting TDA to hypergraphs including the the analysis of Betti numbers of hypergraphs, the cohomology of hypergraphs, and the embedded homology of hypergraphs (Chung and Graham, 1992;Emtander, 2009;Bressan et al, 2016).…”

Section: Velazquez 2005) Defined Asmentioning

confidence: 99%

Topological Distances Between Brain Networks

Chung

Lee

Solo

et al. 2017

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Many existing brain network distances are based on matrix norms. The element-wise differences may fail to capture underlying topological differences. Further, matrix norms are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to develop network distances that recognize topology. In this paper, we introduce Gromov-Hausdorff (GH) and Kolmogorov-Smirnov (KS) distances. GH-distance is often used in persistent homology based brain network models. The superior performance of KS-distance is contrasted against matrix norms and GH-distance in random network simulations with the ground truths. The KS-distance is then applied in characterizing the multimodal MRI and DTI study of maltreated children.

show abstract

The embedded homology of hypergraphs and applications

Cited by 36 publications

References 15 publications

Persistent spectral graph

Persistent spectral graph

Persistent spectral–based machine learning (PerSpect ML) for protein-ligand binding affinity prediction

Topological Distances Between Brain Networks

Contact Info

Product

Resources

About